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Gibbs phenomenon and ringing in square waves: causality? 
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#1
Nov313, 02:23 AM

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A seemingly good way to understand the overshoot and decay (ringing) of a square wave on a scope is that it is the result of bandwidth limiting. In that case, the Fourier series of a square wave
[tex]\Pi(t) = \frac{1}{2 \pi} \sum_{n=\infty}^\infty \frac{\sin(n \omega/2)}{n \omega/2} \exp(i n \omega t)[/tex] shows the wellknown overshoot when n is only taken to a small number, such as 30, terms. It's obtained by Fourier transforming the rect function [itex]\Pi(t)[/itex] to get [itex]\mathrm{sinc}(\omega)[/itex], then transforming back to the time domain, except you form a Riemann sum instead of an integral and take finite terms. But this calculation doesn't reproduce what you really see on a scope, because it appears to be noncausal. That is, before the machine that drives the signal starts to apply voltage (at [itex]t=5[/itex] in the plot), this curve is already ringing. There's a few references to the causality being related to the Kramer'sKronig relations, especially in the "Understanding the KramersKronig Relation Using A Pictorial Proof" white paper availble online (here). In that they suggest forming a causal function, where it is zero for all [itex]t < 0[/itex], which we could do by translating the rectangle function so that its rise is at [itex]t=0[/itex]. Then form even and odd functions out of the signal. When you do this, the odd function is the same as the even function times [itex]\mathrm{sgn}[/itex]. However, this seems to be of little help, particularly because finding the Fourier transform of the odd part [tex] h_o = \mathrm{sgn}(t) h_e = (1/2) \mathrm{sgn}(t) \Pi(t) [/tex] involves a convolution of [itex]\mathrm{sinc}[/itex] and [itex]1/i \omega[/itex], which is quite difficult (Wolfram integrator gives an answer that is not pure imaginary, which shouldn't be true). What am I missing here? Is there a way to present the real signal you see on a scope trace as an instance of the Gibbs phenomenon? If so, is this on the right track? Does anyone know of a treatment of this problem? Thanks! 


#2
Nov313, 05:04 AM

P: 3,106

I've not seen real scope traces that look like your "ideal". They normally look like these from the web.. http://www.craigsarea.com/images/good_gate.jpg http://www.chatzones.co.uk/discus/me.../1218/2480.jpg http://i.stack.imgur.com/Q0uig.png Dual beam scope traces can sometimes be misleading because of the way sync and chop/sweep works. The input and output wave form can be shifted so they appear to be non causal. It's not always obvious what feature of an input wave caused what feature in the output. 


#3
Nov313, 08:40 AM

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For low frequency waveforms it is sometimes possible to 'Chop' between inputs and that will also give the right relative timing (but the chop frequency needs to be a lot higher than the signal repetition frequency. The actual delay through a filter will be at least as long as the impulse response of the filter, looked at in the time domain.You can never have the cart arrive before the horse. 


#4
Nov313, 01:28 PM

P: 10

Gibbs phenomenon and ringing in square waves: causality?
Put simply, how can you calculate the trace in the third pic that you linked to? 


#5
Nov413, 02:32 AM

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In a simple digital filter the output at time t is a function of samples either side of t. Clearly a real world, real time filter can't be forward looking (noncausal) so a delay has to be introduced. That delay modifies the response and makes the output appear causal. Google found this article but it's not a great answer/explanation. http://paulbourke.net/miscellaneous/filter/ As I said it's been awhile. Perhaps someone has a better explanation. 


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Nov413, 03:07 AM

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#7
Nov413, 10:43 AM

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I think the basic issue here is that in any realworld circuit that acts as a filter, the time delays (or phase changes) are different for each component of the signal.
It is possible to design digital FIR (finite impulse response) filters that don't introduce relative phase differences, but the filter introduces a finite time delay between the input and the output. That would shift the "approx" trace in the OP's plot to the right, and preserve causality. Of course if such a filter is used "off line", i.e. not for real time processing, there is no reason why it has to be causal, and the output can be timeshifted to look exactly like the OP's plots. Real world analog circuits always introduce frequencydependent time shifts. Even a simple "piece of wire" acts like an LCR circuit, not like an ideal resistor. 


#8
Nov413, 12:31 PM

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You need to be careful about what you mean by 'causal'. For a start, to discuss causality here, you can only talk in terms of a single pulse, in time and not a repeating pulse (what you normally see on a 'scope. The Fourier transform of a repeated wave shape involves sinusoids that are in existence for infinite time (strictly). The ringing that looks as if it starts 'before' the pulse 'was always there' before the pulse, even for the original rectangular pulse. It was just being modified by a load of other sinusoids so that the Sum was zero, at all times, except during the pulse. If you filter off the higher harmonics, the cancellation is incomplete so that 'ringing frequency' becomes visible. This is nothing to do with causality.
Yet again, there is no paradox. 


#9
Nov513, 01:58 AM

P: 10

I've pored over a few treatments of the problem in available texts at the library. Abdul Jerri has a book on the Gibbs phenomenon, and discusses the nonconvergence of the series even in the large limit (point convergence is guaranteed, but the curves do not have this feature). He points out that as the filter window opens, the overshoot does not go away, it simply moves closer to the point discontinuity, remaining at about 9%. All of his plots (and there are several) demonstrating overshoot of the sgn(t) function begin at t=0, and so he elides the question.
Something else I noticed in my reading (do not have the reference ATM) was the technical point that even without a true step discontinuity, the Gibbs phenomenon still rears its head. This is fortunate, since I've yet to find a 0srisetime generator. The text then introduces the Hilbert kernel, purportedly relevant, but drops it after just a few pages. I guess my inherent prejudice is toward the observations I've made in my work. A 900 V pulser that I work with daily clearly shows this effect of overshoot, and never shows any such behavior before the pulse begins (the duty cycle is quite low). This is simple to understand from a circuit pointofview; a TTL pulse turns on an FET on the rising edge, and a DC supply quickly energizes the output. Until this TTL pulse is sent, the system would have no knowledge that something is about to happen. Perhaps my usage of "causality" is imprecise. 


#10
Nov513, 10:42 AM

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